The Hermite Differential Equation is a second-order linear differential equation commonly expressed as y'' - 2xy' + 2ny = 0 , where n is a non-negative integer. It arises in various fields, including physics and engineering, particularly in quantum mechanics and probability theory. Solutions to the Hermite Differential Equation are known as Hermite polynomials, which are orthogonal polynomials that play a significant role in the theory of orthogonal functions. These polynomials are used in applications such as solving the quantum harmonic oscillator problem and in numerical analysis.